General bosonic σ models and string effective actions

ANNALS

OF PHYSICS

200, 148 (1990)

General Bosonic o Models and String Effective Actions H. OSBORN Department

of Applied Silver Street,

Mathematics Cambridge,

and Theoretical United Kingdom

Physics,

Received September 28, 1989

The renormalization of general bosonic two dimensional c models including torsion and allowing for the definition of finite local dimension 2 composite operators and their products is carefully discussed. This framework allows an extension of the Curci-Paffuti relation to be simply derived and also shows how the conditions for the vanishing of the trace of the energy momentum tensor may be derived from an action to all orders of perturbation theory. This derivation of an action for the massless modes of the closed bosonic string is related to Zamolodchikov’s c-theorem. Detailed calculations are undertaken to two loops and the action then constructed is verified to coincide with a special case of previous treatments. The analysis is based on dimensional regularisation. Special attention is payed to the treatment of the anti-symmetric elrv tensor. The main discussion assumes that the d-dimensional regularising space posesses an almost complex structure, although an alternative treatment with E,,” regarded as restricted to just two dimensions is also considered. The renormalised action for the u model is shown to be compatible with the preservation of gauge invariance when the original action includes gauge fields and satisfies an appropriate gauge symmetry. The general analysis is applied to the special case of the u model with fields on squashed S3. ‘c 1990 Academic

Press, Inc.

1. INTRODUCTION There is now considerable literature [l-5] devoted to the relationship between non linear o models in two dimensions and string theories. This interest depends on the essential observation that conformally invariant two-dimensional field theories may be used to construct consistent string theories for appropriate values of the Virasoro central charge. A general non linear c model [6] involves fields which are coordinates # for some target manifold M, or are sections of some vector bundle over M, depending on points, with coordinates xP, belonging to some twodimensional world sheet E2. For closed strings, as considered here, E, is without boundary and is tacitly assumed to have the topology of the sphere S*, therefore neglecting string loops. If an arbitrary two-dimensional field theory is regularised so as to preserve invariance under diffeomorphisms on E2 then generically there is no Weyl symmetry under local resealings of the metric yrv on E2, even if the model has only massless fields and possesses Weyl invariance classically. A Weyl anomaly is 04lO3-4916/90 $7.50 Copyright 0 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

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manifested by a non vanishing trace for the energy momentum tensor 0. The conditions for 0 = 0 may then be interpreted, in appropriate contexts, as corresponding to string equations of motion. For a classical conformally invariant theory, where for each independent monomial Co,constructed from the basic fields with dimension two there is an associated coupling A”, in the Lagrangian L then in simple models 0 = C, /?Y!&, where fi” is the /? function for the coupling Aa. In this case the conditions for a conformal field theory are identified with the equations pa = 0, which can alternatively be identified with the requirement of nilpotency of a BRST charge [7]. It is then non trivial, as shown by Zamolodchikov [S], that there exists an action Z(n), related to the Virasoro central charge, such that

for a suitable positive definite symmetric tensor Gap(i) on the space of couplings. In general non linear r~ models the essential couplings n’(d) are arbitrary tensor fields, of appropriate type, on M and the expression for the trace of the energy momentum tensor is modified to 0 = C, B”Oa, where B” are related to the conventional /?” by diffeomorphisms on M and also by the action of any local gauge symmetries which acting on the couplings leave the action invariant. The modifications required in going from p” to B” are related to operator mixing effects [9, lo] when 0 is defined as a finite local composite operator and are necessary to ensure that B”, unlike b”, is independent of ambiguities in defining bare couplings A”” due to the action of diffeomorphisms and local gauge symmetries. The crucial equations for conformal invariance are then B” =O. It is not entirely straightforward to apply Zamolodchikov’s results in this context and derive an action for these equations [ 11-J. Nevertheless there have been many perturbative calculations of j? functions, for bosonic 0 models see [ 12-141, which have been used to construct by hand actions for their vanishing or to reproduce the low energy expansion of the closed string S-matrix [13, 15-171. An important independent result to all orders for non linear CTmodels was obtained by Curci and Paffuti [ 181 who showed that if B@ corresponds to the dilaton field @, a scalar 0~ M coupled in L to the two-dimensional scalar curvature 6% on .5?, then this becomes a constant when the remaining B’s, such as for the metric g, on M, B$, are zero. This is necessary for consistency since B@ is then proportional to the central charge of the Virasoro algebra in the associated conformal field theory [ 19, 203 and has allowed B@ to be determined from purely flat space calculations

WI. In order to define local compoiite operators in non linear C-Jmodels by functional differentiation we previously extended the usual treatment to include couplings nE(#, x) with an arbitrary dependence on x [22]. The renormalisation analysis, based on assuming dimensional resularisation for the simplest bosonic CTmodel, with couplings the metric g, and dilaton @, is modified by the necessity of additional counterterms depending on a;,? = i3,AI m. This procedure allowed the

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construction of a well defined finite expression for 0 and various consistency conditions followed. One led to an alternative derivation of the Curci-Paffuti relation while subsequently another was used to construct an action, to all orders, for the equations B$= 0, B@ = 0, in terms of i?@’ which is linearly related in a calculable fashion to B*, Bg, at lowest order B@= B@ - 4 g”B$ [23]. The significance of these critical equations may be seen more simply by considering the actions of anomalous Weyl transformations. Let (1.2)

be the generator of Weyl resealings, so that A: y”” = 20~~“’ for arbitrary a(x). h is here the tachyon field which is just a scalar on M, with the basic Lagrangian L containing a term -h, and is here included in Ar so that classically A: S= 0, S=jd%&L, when d= 2 and @= 0. If S, denotes the regularised action including all necessary counterterms ensuring a finite theory for arbitrary A(& X) then A:S,=A$S,+O(E),

dq=jd’r&B.&,

(1.3) AfS,

a Jddx&c$Q-2[h]),

using the equation of motion and where O(E) represents terms which are finite operators with coefficients vanishing as E= 2 - d -+ 0 and so may legitimately be discarded ([h] denotes the finite local composite operator corresponding to h). It is straightforward to show that

CA:, A:,1 = O(E),

(1.4)

[A:,

(1.5)

and hence as a consequence of AT]

=0

we must have [A;,

A;,] S, =O.

(1.6)

This equation is not trivial since the B’s contain derivatives ahA, for instance + . . . . The requirement (1.6) implies consistency conditions which are just the Curci-Paffuti relation and a formula for aLB@ in terms of arbitrary 8hA which was used by us earlier to obtain an action I(g, @) (these relations are not strictly independent in that the second also implies the first). In general we show later that [At, At,] is equal to a generator of gauge transformations on S,, involving a vector field F, a (J,aa’ - c a,,~‘) Be. Bh a V’Q

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Alternatively since S,(y) defines a finite theory for an arbitrary metric y@”it may be made trivially invariant under Weyl resealings of the metric by letting Yfl” + 7“” = yp”y”‘, y = det yrv, Clearly

However, in this case there is now an anomaly under diffeomorphisms If

6xp = t”(x).

(1.8)

so that for d= 2, @= 0, df S(j) = 0 then we can show in the same way as (1.3) after using the equations of motion A; s,(g) = ; A& s&q + O(E).

(l-9)

Since now (1.10) it is clear that (1.6) is also a necessary consistency condition and that from (1.9) the same equations follow for a vanishing diffeomorphism anomaly as hitherto for the Weyl anomaly [24]. The intention in this paper is to extend the previous discussion to general renormalisable bosonic c models with couplings, the metric tensor g,, anti-symmetric tensor b, and dilaton @. There is an immediate problem in that the b, coupling also involves the two-dimensional anti-symmetric symbol E”” whose treatment using dimensional regularisation is problematic. In two dimensions, with ePYdefined as a tensor rather than tensor density, there is the relation pJEW = y”oy”P _ yPPyv~.

(1.11)

Attempts to apply this equation, even with an additional overall d dependent factor on the rhs, for d # 2 lead to inconsistencies. ’ Although other regularisations may be used, one consistent approach, within dimensional regularisation, is to treat l as strictly two-dimensional. This has been applied to r~ models [26], in particular for the Wess-Zumino-Witten model when M is a group manifold, and simplicity with is discussed briefly here. However, the essential calculational

f(d)

(d-

’ From 2)(d-

(l.ll), +E~‘P =0 and contracting 1) l p = 0. For an extensive list of references

l

with elrv using pv&’ = (& on this problem see [25].

l)fa;

gives

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dimensional regularisation is lost, the natural tangent space group is reduced from and y”y = (.y^“‘,y”“). It is necessary to O(d) to O(d-2)x O(2) with s”,=sx8, introduce additional “evanescent” couplings fi [28], corresponding to terms in L which would vanish for d = 2, since the counterterms involving .Ypyare no longer the same as those containing y”‘. Physical results are independent of such extra parameters A, which are artefacts of the regularisation prescription, at the expense of a redefinition of the couplings ;I [29] but this necessitates further calculation. Alternatively (1.11) can be discarded in the regularised theory but instead we may attempt to extend to arbitrary d the complex structure defined on .?* by l p’\‘. Hence, as is possible for arbitrary d = 2n, we assume the existence of tensor EP” with the properties

corresponding to an almost complex structure on the d-dimensional extended space .5d [30]. In this case the tangent space group is U(n) and there is no appreciable loss of calculational facility since it is unnecessary to assume that invariance under reparameterisations is restricted only to those preserving the structure ‘d-’ z - 2x z2. Nevertheless as EJ” is not covariantly constant for general d additional counterterms arise. In particular it is possible to construct a vector 2p, vanishing if d = 2, from EP” and terms in L containing L$, play a critical role in ensuring that the gauge invariance under 6b, = aiwj- 8,~~ is preserved for general d and is thus a well defined property of S, to all orders. In this case although the evanescent couplings are essential for renormalisability and gauge invariance they decouple from physical results without any complication. It should be noted that in other calculations employing dimensional regularisation evanescent operators have also had a crucial and desirable purpose [ 311. The details of this analysis are contained in the next section and a finite well defined gauge invariant expression for the trace of the energy momentum tensor is obtained. In Section 3 the Curci-Paffuti relation and the additional relations necessary for the construction of an action are obtained. These are also shown to be a consequence of (1.6). A general formula for the action Z(g, b, @) is constructed up to a problem of showing the existence of an appropriate measure on M (this can be defined through its variation but this has yet to be shown to be integrable). Section 4 contains detailed calculations up to two loops of the additional pieces in the b functions involving the explicit x dependence of the couplings and discussed earlier in Section 2. Various relations are verified and used to construct the three loop contribution to B@ independent of @. This result is in agreement with previous work .and is then employed to construct an action.in accord with general results of Section 3. Section 5 discusses the application of the general formulae to gauged (T models and also to the particular example of the (r model on squashed S3 when there are only three couplings. The appendices contain details of supplementary calculations. Appendix A discusses the alternative treatment of l fl” as strictly two-dimensional which, while rigorous as a renormalisation procedure, creates

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difficulties both in terms of ease of calculation and also proving the general results to all orders on which this paper depends. Appendix B amplifies the above remarks on the diffeomorphism anomaly and Appendix C contains detailed formulae for determining divergences with arbitrary curved backgrounds on which the two loop calculations in Section 4 are based.

2. RENORMALISATION AND COMPOSITE OPERATORS IN 0 MODELS WITH TORSION

The general renormalisable

bosonic 0 model is described by a Lagrangian

L(A; 4) = +ygo a,#; a,bj+

~i?b,

a,# a,@

+ $a’%?@+ ypv ap~iv’yi - h,

(2.1)

with fields &(x)EM, the target space of dimension D and with metric g,, where x E E2 a two-dimensional Riemannian space with metric yrv and scalar curvature 9. P is the usual anti-symmetric symbol with el’ = l/j’;;. I denotes the set of basic couplings in the model, A= (gg, buy @, Vpi, h)

(2.2)

with components arbitrary tensor fields, of the appropriate type, on M subject only to g, = gji, b, = -bji and g, positive definite. g,, b,, and @ correspond to the massless graviton, anti-symmetric tensor field and dilaton in the closed bosonic string while VPi, h are introduced to assist in the definition of composite operators; h can also be regarded as representing the tachyon. For some purposes it is useful to define tv=g,+b,.

(2.3)

Since local composite operators are to be defined by functional differentiation of the action, each coupling is allowed to have an arbitrary dependence on both q5and x

WI, 44, xl.

(2.4)

The action formed from L possesses a crucial gauge invariance under 6b, =

aiwj - ajwi,

6h = -V;F’,

Svpi= -iiz,"a~wi+aiFp, 6L = iP

a,twj a,dj) + V,F”,

(2.5)

for arbitrary ~~(4, x), F,(qS, x) where ah, denotes the derivative acting on x at constant 4. This invariance depends critically on V, &‘” = 0 in two dimensions.

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With dimensional regularisation we extend E, + Ed with d = 2 - E. In this case P is no longer covariantly constant but we may assume the properties ( 1.i 2) corresponding to 5, being an almost complex manifold, and define a vector Li?== E,P vi E,! We may then introduce new renormalisable i( I; 4) = yp” d, @S, ki - W’

so that if, in addition

(2.6)

evanescent couplings R = (k,, Ei, 2”) by

ap~22J, - I;,

t=L+i,

(2.7)

to (2.5), &, = -wi,

sz = V,‘( ie”’ a,, f$’ wj + F”),

(2.8 1

then the gauge invariance is preserved for general d. h includes possible couplings, given by scalars Rh on M, to tensors on Ed formed from 6” which vanish when d= 2. For renormalisability it is sufficient to restrict these to just vector and scalar monomials constructed from 6‘” of scale dimension two for which an explicit basis is given by P, P9, and E,, V2eGv, 4 = 9 - ~E““E”~%‘,,,.~‘, Instead of adding i to L it is also possible to redefine V’rir h by VP, -+ vpj = Vpj -t Spk, - iepY$,,~r,

h-+Tz=h+it,

so that l= L(l), I= (gv, b,, CD, VP;, h”). It is useful to define invariants under the gauge transformations Assuming 2H,

= (dt&

= a,b,+

(2.9)

(2.5) and (2.8).

ajbki+ &b,,,

(2.10)

(dvp),=ajv,j-a,v,,i,

then H, 9 A,,=

- (dV,),-

iQ:b,,,

hi=&h+V”‘&,,

(di;),+b,j,

(2.11 )

are unchanged under (2.5) and (2.8). The initial Lagrangian is also invariant under complex conjugation combined with b,+ -b,. This operation, denoted by a bar so that for instance from (2.3) i, = g,- b,, is used frequently henceforth and the corresponding symmetry is always assumed to be explicit. It is also very convenient to use a complex basis for vectors on Ed defined by ip“ = $(e,pe,” - e,@e,‘),

y”” = ~(enpeeY + eapeaY),

egPepp= 26,,,

eaPeg,,= 0,

(2.12)

for CI,Cc= 1, .... n where d = 2n. Thus vtii = ear VP, = Vsi + Z&,Ki,

K,=k,+&.

(2.13)

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Using dimensional

regularisation

then the bare action S, is defined so that

I 441 e -.wx; 0 gives a finite perturbative

(2.14)

measure where

(2.15) ?‘=(g;,b;,@“,

~i,~‘=)=p---e

I+

c X:&C” , > n=l

for suitable 1” and also d,(& E) containing poles in E. The structure of the counterterms is strongly constrained by power counting. If (2.16) then, as will be shown in more detail in the Section 4, it is sufficient to assume that t; = p-“(tv+

T,),

Vi = p -“( Fpi + liXp - ir5,“lT;Xv + aiF,“), P=p-~(h+uihi+Z~+ Y = &(yp”Xp

cc,-.(Z@+$Y),

Y-V’fiF”l),

q . X, + iPXp

(2.17)

.4 . Xv) + Y’,

z=l+t+ai,

Di=ui,

!P=Y.

Consistent with the assumption of minimal subtraction T,, Zi, T, o’, q, ij, Y’, Y contain only poles in E and are constructed just from g, and H,. Apart from pieces involving the last component of XP in (2.16) li, & are operators acting on the appropriate tensor fields on A4 while q, 4 give corresponding local quadratic forms, qT = q, 4’ = -4. Y’ incorporates any additional necessary counterterms involving E,, V’E~” or Z@and which contribute to ho. v: may be decomposed into Vi, k”, E” as in (2.9). S, is independent of F,“, since it appears as a total divergence VPF,” in L,, but is here chosen in order to ensure that L,, rather than just S,, is form invariant under renormalisation and hence

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Apart from restricting the choice of do this condition

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requires

F; = u’~~~

(2.19)

and therefore h” = p -“(.a - a;“‘p

+ Y).

(2.20)

With F,” given by (2.19), A” depends on a:ll only though aktii. Since, with h,, A,,,, given by (2.1 l), (2.16) (dh), = -V’fl(AIPii there is a potential ambiguity in choosing ui in (2.17) and (2.19) although oiai is uniquely prescribed. Also since (dAP),,= epY2(&, H, - L&H,,,) the basis chosen for XP is redundant but in explicit calculations later the structure assumed in (2.16) is convenient. In general the form assumed in (2.17) guarantees that the gauge invariance under (2.13) is maintained at all stages of the renormalisation process. As usual we define b functions by (2.21)

with pi, = PQ + @ = /?:I. In the standard fashion the /I functions are given by the residues of the simple E poles in ,I”, so that at L loops BCL)= LAycL’, while higher order poles are determined by (2.21). With the b functions so defined, (2.18) gives (2.22)

Using

(2.23b) then the assumed structure (2.17), (2.19), and (2.20) for 1.” leads, in terms of the complex basis given by (1.12), to

p~=Ojsz-a,(uit,),

fl~=o,xz-ai(zW,),

(2.24) pxm+$9. x = b&,

a; g,, axijk,

am,,= -(da,),+

41,

a:b,-S2,b,,

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where xT = X and from (2.23b) 5 may be expanded in the same basis of scalars vanishing when E -+ 0 as Y’. As in (2.13), /3,“,= /I; + Z?apF and /?‘= /I” +/I”. Since S, has been constructed to give a finite theory for arbitrary couplings I(& x) it is possible to define finite local cornpositive operators, with the same structure as in the initial Lagrangian (2.1), by2

for any p = (p$, pf;, p4, pLi, p”) with pb = p$ + p$. The additional total derivative terms arising from the dependence of L, on alg$, dhb, are essential for finiteness. From the invariance under the gauge transformations (2.5) and (2.8) it is straightforward to obtain

Since ?$ = O(E) then whenever it appears in conjunction with a finite operator, as in (2.26b), it may be set to zero to obtain strictly two-dimensional relations. However, terms containing =!ZPin L, cannot be discarded since here the bare couplings contain poles in E. There are also important relations among the composite operators defined above as a consequence of the equations of motion. These may be derived from invariance under diffeomorphisms when S#’ = -vi, Sl = PUx where 9” denotes the Lie derivative corresponding to a vector field ui. Since this symmetry is preserved by dimensional regularisation it is assumed to give

sqj; = -vi,

(2.27)

Hence

The lhs of (2.28a) is a finite operator since the second term is just of the form prescribed by (2.25), while E(u) = -2rca’v’SS,/S&, which vanishes on insertion * For that

a more

precise

definition

of the functional

p’.alar=pp.a/ag+pb.a/?b.

and partial

derivatives

used here see Ref. [23].

Note

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into the functional integral (2.16) as the measure is invariant under diffeomorphism with dimensional regularisation, is the equation of motion operator. This CE given by (2.28b) must also be a finite operator up to terms which would vanish on the rhs of (2.28a). With minimal subtraction this requires that C; = [dp@g,o’-

ie”’ d,,qYb,o’+

@I’]

+V,.(i&“‘X#)

- P’X,

S:r’,

(2.29)

with Xi = - 8, containing only poles in E. From this explicit form for (I’: it is then possible to use (2.26a, b) to recast (2.28a) as v,,[aygj’o’]

- [a~qPg, a;$‘]

iI,- = L&l + (0, (du~)i,, 0, -ai(

= (Vi v/+V, ui, 2H,,,v”,

Ppju~)-

= [L(X,)]

-E(u),

(2.30)

ie,ya:, h,;- 2&b,) v’, v; pt:y

vi a,@, &pi,

U’, = h,u ‘.

hid),

Consistency between (2.28b) and (2.29) requires the relation a,,&) g,Tu’- i.5,” a,q5’,b,~ui+

V$d

or with the complex basis (2.12) this gives two independent

conditions

ui =

+ p-“Z( P,,u’) - (a;x; - 9Ji)

8.

f,k

vk,

(2.31a)

(2.31b)

Using

where Vi, = ~(“2: result

V, + O(aht),

then (2.31a) is equivalent

@lj= oiz, - aicuJv, - Fp’).

to just the single finite

Fj = - F,

(2.32)

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with

a ava, = ( - (du),,+ 2H,ok, .-+Tnt.-

=%gij,d%:Hqk,O),

(2.33)

and assuming, since aiZ = Z,r” a, Xjv’=

-p--“Z(U’b,,t?+

(2.34)

Fjv’).

This result, with Fj finite and constructed from g, and H,, is necessary in order to achieve the gauge invariant expression shown in (2.32). Similarly E-fl’ .a/& -fi:.a/iYV, may be applied to (2.31b) and then (2.32), (2.34) used to obtain oiO,Xx=

U’(A”,,vj)--a&U’oi-Xa

.x .2” + (a~Fi-9mFj)

vi,

(2.35)

assuming that for St, = g, tii, 6( UjVj) = v’ai ( UjVj) - UjP” Vi. In both relations (2.32) and (2.35) all expressions are determined in terms of the generalised /? functions in (2.24) except for Fj= -Fj. Just as in (2.25) a finite energy momentum tensor is given by

ss

T,,(x) = 47ccX’0 Syyx)’ where the variation

of E”” is computed

using

SE/ = 4 6ypoP + fell0 Sy”‘, which is compatible

(2.37)

with (1.12). Under Weyl resealing iy” = 2oyfi’“,

~22~ = --E awa,

&&=-do&

(2.38)

si4T= 209 + 2( 1 - UE) v2t7,

with a an essentially unconstrained

parameter reflecting the fact that an arbitrary into the definition of &?, conventionally a = 1. Using (2.38) and also assuming 6 Y’ = 2oY’ - EC V 2~ + C’P‘ a,a and similarly sfi = 2ah - t V 2~ + F a,a then for the trace we obtain d dependent factor may be introduced

ypvTp, = EL, + 2pPZh

+v,

+ ; a’( 1 - ae) V ‘Go (2.39)

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(under differentiation L??~is taken to be independent of any scalar monomials appearing in I”). From (2.22), and since [F] = p-“ZF for any scalar F, this becomes s,+Zl~,+~l’(l-a&)V’CII+V,,P,

yL(‘Tp, = 27d (B.h+B”-$)

(2.40) p=J~+p,

P =V”[E]

+ [?“I,

where s/1 = O(E) and

and we may impose JP = J” implying A’= - K Since all other terms in (2.40) are well defined composite operators then so must the current Jp be als a finite operator, for suitable X, such that J, = [&q@i]

Combining

(2.42 )

+ [U,].

(2.41) and (2.42) then gives 2

a B“I.--i:-&a $.-.-( av,, aa:t

cL 1

L,+p-“ZU,

(2.43) In turn this may be decomposed as in (2.31a, b) leading to

=P

~&a,((i-aE)~+x+d7

(2.44)

and also the same expression as (2.43) with L, + -h”“, 8, + ai. To obtain equivalent finite form we apply E- a’ . a/at to (2.44) which produces

=ai((i-~&)e+a+d) assuming (E-B’. d/&)x= Ze, with e= --a, Z(6, 6’) which follows from (2.23b). Writing

si = s, + EIq,

-3. ,.&.+)‘-&+E&)Jy,=ig+l.

an

(2.45) and also (&-fit

.d/at)(C,

C’) =

12.46)

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where

9 = (0, g,, H,, 11,

39= WS), + B& a$ fWbhjk, 01,

(2.47)

then (2.45) may be decomposed into O(E) and O(1) pieces:

si=oi9+ai(-d+s),

(2.48a) (2.48b)

The first equation determines Si but this is more conveniently although equivalently obtained directly from (2.44) in the form Si=(li+7;:),9+di(-aY’,+C,),

(2.49)

with !PI, ... denoting the residue of the simple E pole in Y, ... . Within the formalism presented here Si is automatically and manifestly gauge invariant, depending only on g,, H,, and, as shown later, zero at one loop. This is a direct consequence of the role of the additional couplings k;, k; since otherwise, in their absence, 9 = (b,, ...) which d estr oys these properties and requires additional counterterms to restore the symmetry [13]. In an analogous fashion the remaining relation contained in (2.43) can also be expressed in an equivalent finite form,

1 a -2 ('i.x-B

a a .-aagt &BY,1 B

u, + ,/+y.-g+Q ( Ir ) =a~((i-~&)o+~+&b)+~,(s'-~). The terms in (2.50) containing an explicit factor of E determine an alternative expression for U, follows directly from (2.43) u, = u’P,+

w&

W~~=al,(-aY,+C,)-~-(q+~),-~~,

(2.50) U,- but, as in (2.49), (2.51)

using (2.23a). Given the freedom in choosing uj and hence via (2.23a) in Uj it is possible to assume W is such that WSti is independent of V, or equivalently WY: = Wu),,

0, 090).

(2.52)

With (2.51) and this result (2.50) becomes

a;(e+a)+d,(6k) 4

P’.7-Q

at

)

(W~~)+~.~.~~-aBUjsj-Uj~j~~.

(2.53)

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The significance of these final formulae (2.48b), (2.53) is that they determine the dependence of 9, occurring in the dilaton /I function, on (b and also g,, H, up to the scalar 0 which may be easily eliminated since 8= 0, ci = -0, Alternatively a relation determining (r may be obtained by applying (si. a/aVij - p^’ . LJ/&Ygt - &a/&&) to both sides of (2.43) and subtracting its conjugate, using L, = L,, C = C, C’ = C’. From (2.46) this gives, after factoring off 2, from the O(E) and O( 1) pieces, FVF = wg,

(2.54a)

WB - Wg = 2(a + U/r,).

(2.54b)

Another crucial relation, which is important subsequently, between W and S follows by applying ( - Ki. a/a V,; + t . a/&3; t + a/Z!J), which acting on XX gives Y in (2.47), to the simple pole terms in (2.31b) and using (2.23a), (2.34) with the definitions (2.49), (2.51) to give (2.55)

S,v’ = W2” + U’v, - F, vi.

The expression (2.40) for the trace of the energy momentum tensor may now be simplified using the equations of motion (2.30) and (2.26a, b). Using = to denote equality up to terms involving finite composite operators with coefficients vanishing as E + 0 which may therefore legitimately be discarded in obtaining physical twodimensional results. J,, *V ca,45,i yT~,. N [L(B)]

Finally,

+ ie,‘[a,@rJ

Si=s,+r,,

+ Cu,],

+ 2[h] + OCR’ V”[@]

+ V,P

= 0 - E(i),

Si=s,-r,, s, = si + ia’ a, @.

(2.56)

setting V,,i, a;t, and hence U, to zero, then 0 = [L(B)]

+ 2[h],

sg=~~+vJj+vjsi, BfJ = pi + 2H,Sk

Bh = Ah,

+ air, - a,r,,

(2.571

A = Q + s”’ 8, = (S’ - U’) 8,.

Unlike the /3 functions the B’s do not have any arbitrariness resulting from the freedom of modifying 4, by a diffeomorphism with consequential changes on ,I^, the variation in /I is compensated by a corresponding change in si. Also h& is potentially arbitrary due to the symmetry (2.5) but the resulting ambiguity in /?i is cancelled in II: by a corresponding variation in r,.

16

H. OSBORN

For subsequent purposes we also need the additional V,,i and i3g tii. In this case

contributions

depending on

(2.58)

3. DERIVATION

OF THE STRING EFFECTIVE ACTION

The essential results of the previous section, in particular the consequences of diffeomorphisms contained in (2.32), (2.33), (2.35) and of scale variations in (2.47), (2.48b), (2.53), are there expressed in terms of the conventional o model fl functions. It is however straightforward to reexpress them in terms of the B functions given by (2.57). From (2.33) and (2.47) it is clear that B” = 93+ gls= (B;, Bfj, ;(dBb),,

0),

(3.1) Y = (b,, g,, H,,

Hence combining

0) = t .&

a

S,.

(2.32) and (2.48b) now gives -ai(e+~+u,s,-cl,jj)=OiB”-P;S’+B’.~Si.

(3.2)

Using

since Si transforms as a vector, then with the definition (3.2) becomes

of B@in (2.57), the relation

-d,(+$B@+cf)=O,B”-B;.S’+B’--$Si,

(3.3) at=a-(gj-Uj)rj-Fj~J=

-5’.

From (2.55) and (2.54b), with (3.1), 2~’ = WB” - WB”.

(3.4)

Hence (3.3) implies ai B* = 0( Bg, Bb),

(3.5)

(3 MODELS

AND

STRING

EFFECTIVE

ACTIONS

and is therefore the direct extension of the Curci-Paffuti relation to D models torsion. This is a necessary consistency condition since for the conformal theory, obtained when B$ = 0, BfJ = 0, 3B@ is the Virasoro central charge and therefore be a constant [19,20,23]. Similarly from (2.53), using (3.1), (2.35) with u’-+s”’ and (2.55) with I?-+ and also the result that for 61= B - /?, S%* = LG$~~ + yaij we obtain

17 with field must &F’

(3.6) The results (3.3) and (3.6) are not independent.

Lg!= uj -g+zJ XJ

If

a a .-+.g"@.-ad&t aap'

uj= t/kvk,

so that from (2.33) 9” Ti = y”, then using (2.58) L@,.B~~ = Bfi vi. Thus applying to (3.6), with W$@ independent of Vgi, gives +'B@+d)=

;

B".Y..X+(S'-(~')(B~~')-B'.~(~~,)

= -v'OiBg + $'B!.,j- B' . za S$,

(3.7)

by virtue of (2.36) and (2.55), which is identical to (3.3). An equivalent expression of (3.3) and (3.6) may be found by using them with (2.58) and (3.4) to write B(Y) .-

6

WY)

%)=6(x,

y)B+B;(x) (3.8)

B(Y). -&B”W=6(x,

YP+~(x)

where (3.9)

18

H. OSBORN

Hence

[

B(Y).-

6 WY)’

+,6(x, -V@

6 -U(x)

B(x).

y)

(

1

aiB@(x).&+vwyx).T& w 6 s@(x).,ho +x-y , (

>

(3.10)

)

or

[A:, @,I = &

(3.11)

AFp is the generator of F, gauge transformations in (2.5) so that (3.11) implies (1.6) which, as discussed in the introduction, is a necessary consistency condition for anomalous Weyl scale transformations. The importance of (3.6) is that it provides a general expression for the variation of 8” with gti, b,. Setting VP to zero, (3.6) gives

(3.12)

As a consequence of (3.12) an action can now be obtained whose variation implies the string equations of motion for the massless modes, B+O,

B;=O,

B@=O,

(3.13)

0 MODELS

corresponding

AND

STRING

EFFECTIVE

19

ACTIONS

to 0 = 0, if a measure J on M can be constructed so that s

(3.14)

d%$J(s”‘- Lqoj=o(B’),

for any vector field vi(~). In this case (3.15) satisfies (3.16)

61= O(Bg, Bb, B@),

by virtue of (3.12) and (3.14). At lowest order, as shown later, the result of one loop calculation

gives s’ = 0 and

s’- Ui=+y(Vi~--vi).

(3.17)

Hence (3.14) is satisfied, with zero rhs, if g = det gl,.

J=JO=&epO,

To arbitrary by

order we may suppose that J= 1 is determined

(3.18) through its variation

6J= -J~~-~J(ware+w6~)+ai(~~‘61)+o(a’),

where the total derivative and O(B’) terms are regarded as determined so that 6J is integrable. To one loop

(3.19) from W, W

so that (3.19) is in accord with (3.18) without the necessity of the additional terms for integrability in (3.19). In general the solution of (3.19) may be taken to be of the form

J= Jo&, H),

J= 1+ O(a’),

(3.21)

where % is a scalar (since the @ variation is obviously integrable as in (3.18) the dependence of B’ on the dilaton field involving a@ may be supposed to be cancelled by the total derivative term and as 6J= 0 for 6b, = (dw), so also J is gauge invariant). Under a diffeomorphism

20

H.

OSBORN

so that from (2.53) and (3.21) 6J=c?,(u’J)=

-$J(?-

U’)uj+a,(JG’~~t)+O(B’).

(3.22)

Thus this implies that J, as determined by (3.19), satisfies (3.14). According to Tseytlin there is a scheme, obtained by a suitable redefinition of the couplings, so that (3.17), (3.20) and hence (3.18) are valid to all orders. Unfortunately at present there is no general justification for demonstrating that 6J as given in (3.19) is integrable. At one loop

x, . x . Tti = ~u~gikg’~(a~g,, a;gkr- &&),

(3.23)

so from (3.12), (3.15), (3.19), (3.20) and (3.23) 1 61 iu’gg=e

-q-p+

. ..).

(3.24)

where the discarded terms correspond to the effect of higher order contributions to K 1. A direct corollary of (3.12) follows by taking 8% cc B”, since P,B = 0 is equivalent to the Curci-Paffuti relation (3.3) with (3.4), giving (3.25)

with d given by (2.56). This is an analogue of Zamolodchikov’s equation for the renormalisation flow of the Virasoro central charge modified by the presence of the differential operator A as a consequence of 8@ being a local scalar on M [23,26]. Equivalently if

denotes the Callan-Symanzik differential operator generating renormalisation mass scale p then, since for any local scalar

Wl=[(L++]

resealings of the (3.27)

21

B MODELS AND STRING EFFECTlVE ACTIONS

(up to finite O(E) terms), (3.25) implies g[jj”]

z.$ [B” .x . B”],

(3.28)

since 8@ is a gauge invariant scalar under diffeomorphisms. This equation plays a central role in the application of the renormalisation group to correlation functions involving the energy momentum tensor [32].

4. ONE AND Two LOOP CALCULATIONS

In order to determine perturbatively the various /? functions of Section 2 for arbitrary A(+, x) we follow conventional background field techniques and calculate at each order the divergent part of the one particle irreducible connected generating functional T(q), depending on a background field cp’(x). After removal of subdivergences I-,,, is a local functional and may be cancelled by appropriate contributions to L, of the form (2.15) and (2.17). To achieve a covariant perturbation expansion the quantum field di is reexpressed in terms of {‘, the tangent vector at cp to the geodesic path dP from cp to 4 for each x separately. If ri, is the usual Christoffel connection constructed from the metric g, then *-i + Iykf$;qJ; = 0, q$

d;(o) = (Pi>

The resulting expansion of the Lagrangian

where Dt is a derivation be calculated by using

c&(O) = ri.

qql,=$K

(4.1 1

may be represented in the form [33]

whose covariant action on any terms occurring in z,, may

D~‘,,...i~(cp)=Vi’i,...I,(~) D; diicpi=Vgti,

4’3

D
D< V, (‘= (Rllkl a,,+

d;r;,)

D;t’=O,

(4.3a)

5”tk.

(4.3b)

Vk denotes the covariant derivative with respect to cpk on tensor fields on M determined by the connection r&, with R>kl the corresponding curvature tensor, and similarly V, acting on such tensor fields and ti has the associated connection given by a, ‘pkr;,. Employing this prescription to calculate D,E( q), setting @= 0 for simplicity, and discarding the total derivative V,(i&‘“b, 5’ a,(p’+ VP(‘) gives the gauge invariant expression (4.4)

22

H.

OSBORN

In the complex basis of (2.12) v’,[i=V;ti, V’,li=Vili correspondingly Vi*, defined in terms of the connections, f+’ kJ.=r’

+H’..

kJ -

with Vi,

Vz,

and

(4.5)

hJ,

so that H plays the role of a torsion tensor, [V;,V,+]

v’=R”,~~~~~H~~,V~V~.

(4.6)

As a consequence of dH = 0 the associated curvature tensors satisfy the identities R&l = RGii,

Vi’ Hjkl=

v+Hjk,=

$Ri;JiG/j 3

;R;+.

(4.7)

The calculation of z,, may be further simplified by including also the torsion term in the action of D, on tensor fields since the initial geodesic equation (4.1) is not affected by torsion. Thus (4.3a) remains valid with D, -+ Df , Vi + V + and (4.3b) is replaced by

apqi=vpT p,

D:

Applying

v:~i=(~+ii~~a,~[-a~r:~)5j5k.

D;

these results it is straightforward 2,

=

g&

(W[’

xv=a,cpk

+

to obtain from E, in (4.4) fi,f=v,
X,[‘~J,

+

&d%,~,

i

(4.8)

$jlp’j(j,

az(pkA”tik+(i;J)

+

4

(4.9)

di’pkaark--(i;j)

-a~(pkgk,a;ri,+tAl,k(iA”,j)k-i;(j;j), where Akki;J=Aki;jfA,iH:.k. techniques when calculating

In order to allow for the application of conventional divergences it is necessary to introduce D-beins,

gi,=e,:e

eniebi

aJ ’

= bab,

(4.10)

5, = eX,

so that e,i~‘,5i=~~5a+Spob4b~ qhap

= ecai a;e,,‘=

spab ta+dpabtb?

sle,,b

=

$eaiebJ

a.y,

(4.11)

- sB,bw

In this basis (4.9) becomes z,=~~5,~~5,+~b5,5b+v,(s~b:b5,4b),

Xab= e,‘eJXv+ pJ

= p

+

(s,sP-V,,Y--

ij aLgikgk,

afpg’i

_

[s$, s~])~~= eUieJXv, is,

In the standard fashion the one loop contribution r(l)=

-Ilndet(-g2+X) 2

(4.12)

alpgij.

to r is given by

w-j--Jd’x&(:W-tr(X)).

(4.13)

cr MODELS

AND

STRING

EFFECTIVE

23

ACTIONS

The divergent part may then be cancelled by taking 1 L(I)=-a’ 0I 2

D

6P-g”X,i-$g,,gkia;g’ka’“‘R”-V”(V~~~j)

(4.14)

1

(

It is then easy to read off, noting that L(cp)=it,d,cp’a,cp’+ ..., the different contributions to 1” and hence determine PC]‘. From (2.17) and (2.24) #I) = la’D 6

p;;l’=g’R;, O)“XE=

$a’( -v-‘A,,+

u”“=

‘,$V’ 2

)

cpll=o

,

’

(4.15a)

gi,g’k a;r:,)

= $a‘(V --‘c,Q - ai+(g’k a&g,)),

(4.15b)

CAL;, = a;, g,, - &,,

along with x”’ which is given by (3.23) and Q”‘= hence C’” = C’“’ = 0, (2.49) gives $I’= 0 I

- 41x’V’. Since Y”” = 0, and (4.16)

while the definition (2.51) leads to (3.20). Under Sv; + A,, cMii + 2 V,?v, so that using v-‘V,?vi=

aj(Vivj)

+ R;

implies that 0 I” is consistent with (2.32) and (4.15a) for FI” =O. Similarly Xx + B, with S, = 0, cxii -+ /?Ji and since VP’R;

it is clear that (2.48) is satisfied with e(*) = _ ia”H’ Writing

H’=

-~&(gjkR;)=$d,H’,

(3.23) alternatively

#I

,

H,,k Hqk,

= 0.

(4.17 I

as x . x (1’. x* = $‘&$;,,

(4.18 )

then the rhs of (2.35) to this order becomes

~a’(V~(Al,i,o~)-a&v’L~,-C~V;vi)=~a’(o’V-’A”,,,+L~~g”a:,f&) in accord with v’O~‘)XE from (4.15b). Also the rhs of (2.53) becomes 1 2 - a’

( 2 I((

R+ .l+v*

(g"a:,gii)-(viv+j-R+")c,i, >

.>

2 (g”a;(H’),+

1

“2 3 (d;H’+

=-ta9 2

for

HekVk&) !&HZ),

24

H. OSBORN

using (4.6) and other standard identities with (Hz),= fore consistent with (4.17) and implies also d’(2) = _ (9)2 5H2.

Hik,Hjk’.

This result is there(4.19)

Ultimately (2.57), using (4.15a) and (4.17), then leads to the well known one loop expressions [3, 51 B$‘)

= a’(Ru-

B@“‘==mcr’

1 2

p(l)=-,,

1 2

(H);

+ &Vi@),

n+ak@ak@-vz@-gH2

B;J”=a’(Vk+Vk@)

H,,

) >

n+akdjake2vQ-R+fH2

(4.20) >

.

At two loops we use an expansion of the basic Green function for the operator -9’ + X based on the Dewitt heat kernel expansion [34] and used earlier for calculations without torsion [22]. For (t’(x) tj(x’)) = G”(x, x’), determined by z, in (4.9) this has the form G” = Goa; + G, uf + G” 9 (4.21) where GO, G, are singular functions, given explicitly in Ref. [22], -.. depending only on the spatial metric yrv and well behaved as d + 2 while G” and its first two derivatives are non singular for x+x’ so that it does not contribute to local divergences. L&X, x’), ay(x, x’) are regular for x - X’ and their coincident limits may be readily computed

The divergent parts of products of GO’s and G,‘s with derivatives may also independently calculated and those relevant for present purposes are given Appendix C. The two loop amplitudes are determined by I?, and I?,. After subtraction subdivergences, or computing the required counterterms from L, and 5, [35], leads to only double poles in E which are irrelevant for computing /? functions. is easily calculated to be

be in of & L,

(4.23)

where C, is also unimportant since it does not lead to divergences. The most singular two loop amplitude arises from the term in 2, with two derivatives, r’2’= a

-22ncr’ 1 H-- H;,,,,E@~P 9 ss I@ x (~,,G”$:~,,Gj~~~Gk”+2~,Gil~:,~,Gi”

Gk+),

(IT MODELS

Subtracting

AND

STRING

EFFECTIVE

25

ACTIONS

subdivergences, using (4.21) and the results of Appendix C, leads to

271a’ r(2). =HiikHI,,,,,a{d$‘ak, u d’v (47c&)2 i 1.i

$(1+E)V2++?

+~“Ei’.,V2C”V-~(4+c)(V,,~‘+~~~~)

>

tid

+~(2-,:)j((H2),Bi’ag$ll-H~i~,H,,”’~~a~la~~:,l) +(l-e)j(H’),a@l

(4.24)

. I

Contributions involving B$ are easily found and hence from (4.22) the two loop single pole part of L, has the form L$’ = (ia’)Z - (@-

(&Bflllk

Bylk

+ ~iek’YHi,kB~c,,lk

&gp$!” + &Q”@‘)

H’-

$2”

z,,H’

+(H2)“(X~+~gk’~;,giki3’~‘gi,)+V’L((H2)i’ + H,, Hklm $;cpi’

Pi,J

+ gikg”( - ib@(H’),

L3i1gk,

+bg’““~“H,~,H,,-t~‘H,,,,H,,,,c~)).,

(4.25)

using integration by parts where appropriate to achieve a form compatible with (2.16), (2.17). In this case there is a non zero contribution to Y’ whose anomalous Weyl scaling properties may be calculated by using under (2.38) ~Y(E~,, V2eP’) = 2ae,,, V”P

- 82” a,,~,

giving C (2) , = 0 and C;‘2’ = _ ($af)2 fH2, in accord with (4.19). Similarly Y”,2’ = - (+a’)’ iH2 so that direct calculation agrees with (4.17). From (4.25) the remaining two loops results, after some simplification with extensive use of identities such as (4.7) are conveniently written as /l:12’ = ($a’)’ Y’n’kjR~,m, ui(l)= -(Iaf)2 2(H2)“V, 2 I’ Oj2’.;T,=

($a’)? j-di((H2)jk

x%. xc”. $YY= ($a’)2 (3dL”l

52’” = (ia’)2 2(H’)” ii:g,)--

V,?,,

Y’r’i dzlk,+ idiH21,j,

dgijk - 2dzk z4,ik (4.26)

+(H2)~c,,,c,i”+a:(H’)“~:g,,+dX(H2)”i!:g, +~~~~~H2-~(~!1,1H’+,1,:H’)), Y/m,/= -2Rkkj+

3R&,w,,, +2(H2)k,

2dg,k, = V : $k/ - V : Cz,/+ 2Hik”‘C.,,,/,

gn,/-‘2(H2)koz

g/f,

26

H. OSBORN

with ctik, given as in (4.15b). The form of the expressions in (4.26) is such that verification of (2.32) and (2.35), with Fj2’ = 0, is quite straightforward since under XU --t 2” then dmjk,-+ -u’R,& by virtue of (4.6). To this order from (2.49) or (2.48a) and (2.51) or the O(E) terms in (2.50) S!2’ = (La’)2 q a I.H2 Y 2 H”2’s-i$=

(;a’)’

((H2)“d;g;,+qd;H2+

which are obviously compatible Bf2’ = #”

q= -&a), &H2&),

(4.27)

with (2.55). Thus from (2.57) By’

+ (;c1’)2 2q ViaiH2,

= ,y

+ (+a’)’ 2qH,, akH 2.

(4.28)

It remains to verify the crucial equations (2.48b), which plays an essential role in the derivation of the Curci-Paffuti relation (3.3), and (2.53) which is necessary for deriving the action. These now determine 8 to three loops and after some calculation give +hJk

+ R,kh

3 i 51 RJk’“Rjkl,,, - 3 (3q + 1) V”Hjk’

=ka921

v,,, Hjk, + ; H4 (4.29)

+ (6q - l)(Rjk”Hjk,HImn

T (H’),&(H’)j&)

,

H4 = H,,,,Hk,“Hik,H””

(note that the parameter q here is twice that of Hull and Townsend [17]). (2.57) and (4.26), (4.27) we get P2)

= (

i a’ 2 (q ajH2 ap + 2(H2)” >

V,a,@) + s dc3),

From

(4.30)

and hence from (3.9) and (4.29) 8@(2)= (;a’)’ +4(H2)” ~(+a’)~ -3(H’)”

{ -2qV2H2+q

aiH2 a,@+ (4--6q)(H2)“Vjaj@

R; - R+mk”R&,,+

;H4}

{-~R~k’mRJk,,,,+Hjk,H,m”Rjk’m+4(H2)’iR~-~H4

(4.31)

(H2),,+(4-6q)(H2)‘iV,a,@-2qV2H2+qaiH2ai@}.

The measure J= J,J, J, = J-g C@ may be constructed by using (3.19) since J'=

1+

+a’(+-q)H2

(4.32)

27

(5 MODELS AND STRING EFFECTIVE ACTIONS

satisfies Jo sJ= - &‘((H2)”

6g,+ q6H2) + ia 8;(JoH’ik hb,,) + &row

6bi,,

in accord with W”’ given by (4.27) and with Bb as in (4.20). The action for the string equations of motion may now be constructed according to the general formula (3.15) to two loops using (4.20) (4.31), and (4.32).

!

-;

RikfmR,,,, + H,knH,,,,“Rik’m +4(H’)”

-3(H2)“(H’)j,+(4-6q)(H2)“Vi8,Q,+

R,,-;

H’

(3T-3q ) 8’H’a.QI )},

with some integration by parts to reduce the number of terms involving field. The parameter q can be eliminated by the following redefinitions

(4.33)

the dilaton

6@= - fct’qH2,

The result (4.33) is a special case of a general class of actions constructed by hand previously after appropriate changes of basis.3 The main constraint on the form of the action constructed here is the restriction on the dependence on the dilaton field.

5. CONCLUSIONS

One of the principal advantages of dimensional regularisation is that it preserves all those symmetries of a quantum field theory except those which are valid only for a particular dimension while the virtue of the treatment of E,, in Section 2 is that it is possible to ensure that the gauge symmetry 6b,= (dw), is maintained. All counterterms therefore depend only on H, = i(db),,. A natural application of the results obtained in Section 2 is to consider a gauge group H and corresponding external gauge field A,, so that the non linear c model is invariant under the gauge transformations

‘Take (H')"V, d, @ --* (Hz)” d, @ S, @-V’H,k,H’k’?, @2,1--t -A, 2y+4, I,=12=I,=~,=~,=~4=145=~b=o, i,=3+3q. 2i.,= -5+3q, 2&= -13+27q or alternatively v= -1, p= -4 with q=O.

b?’ H’?, @ and in Eq. (62) of Ref. 1171 p2=-2, p,=-2+3q. p*=f, p,= I, in Eq.(18) of Ref. [16] D-r -@, ;= I,

28

H. OSBORN

k: are a set of vector fields on M such that (5.2)

with f& the totally anti-symmetric structure constants of H. In order to ensure gauge invariance it is usually sufficient to introduce covariant derivatives so that a, +

a,+A,,k;.

~,df=

The metric term in the c model Lagrangian

(5.3)

(2.1) is then invariant

=%g,=o,

if (5.4)

with Ya denoting the Lie derivative associated to k:, so that k: are Killing vectors for the metric g,. It is however not appropriate to introduce covariant derivatives into the torsion term and demand Tabij = 0 but instead to impose only the requirement ga H,, = 0.

(5.5)

In this case the gauged non linear CJmodel is assumed to be of the form, neglecting any curvature terms, L=fy”‘g,D,~‘D,~++~“‘(fb,a,~ia,~--A,,CYIay~i+~d~bAraA”b),

with now g,, b, depending only on #EM we may write, at least locally on M,

(and not on x as well). By virtue of (5.5)

2H,k;

= - (dc,),,

for some vector field cai = -C,; invariant gauge variations on 4 in (5.1) W“~b,

a#

(5.6)

(5.7)

under 6b, = (dw),.

a,y = iP(a,(A,wJi

a#+

a,n,c,i

Since then under the a,,#),

wai = - biikj, - cai,

the cancellation of the a,A a,4 terms in the gauge variation of (5.6) is ensured after discarding a total derivative. The remaining conditions are easily seen to be -%a

cbi

=

fabc

cci

3

cclik; = dab = - dbo,

and also

(5.8a) (5.8b)

0 MODELS

AND

STRING

EFFECTIVE

ACTIONS

29

which is in fact a consequence of (5.8a, b) since (5.2) can be alternatively written as Toki = fabckf,.” It is clear that for the gauge symmetry to be valid, invariance under 6h, = (d~)~ is crucial. The gauge invariant Lagrangian (5.6) is a special case of (2.1) obtained by taking V,i= ( -g,k’,

+

c,;)

h= - $(g;,k:k’,

A,,,

+ dub) A,,A,b.

(5.10)

It is a significant consistency check to show that the various relations above are preserved under renormalisation. To show this we may use (2.17) (2.19) and (2.31a), (2.33) to write in this case, neglecting p--E factors, v:,= t,;k’,=

(5.11)

V,i+(zi-l,)xti+a,(u’v~,), t,k’,+(l;-l;)&+&(ui(t,,kk,)-X,kj,),

where, since 8; tii = 0 and using (5.4), (5.7),

-x,=(-VW,, 0,0,01, A = (- W,), +(dc,),,0,0,O),

cfti= -jaA,,.

(5.12)

From (5.10), (5.11) and (5.12) t;ki,A,,=tiiki,A,,+

Vtii-

X: = w’(bikkk,

I’;,-a,x;A,,,

(5.13)

+ c,,) - X,k; = -Xl.

Clearly (5.13 ) requires VEi = ( - g,;k’, + c;;) A,,,

C&,= -cz,,

(5.14)

where b;k:

= b,k;

+ c,, - c:;+ ?,X:

(5.15)

Since (du,),=

-Sab,+2H,k;,

and similarly for b, + b,;, H, --* H$ then (5.15) and (5.7) imply 2H$k;

ua,=b,k:,

and also &(b,y - b,j) = 0 from (5.4), (5.5 ), = -(dc,“),.

4 The usual solution [36, 371 of these equations in the gauged Wess-Zumino-Witten obtained for g(b) some representation of H with generators 1,. It,, thl =.foh,1, and if g-l&= dgg-‘= I,?,, d&, e,,e,‘=P,iZ,‘= 6,,, gt,g-‘= thRhrr then kf,=e,‘-P,‘. co, = q(e,,+?,,,), rl(fL - L) when 2H,,k = rlfos eorehieck.

(5.16) model is t,,e,,, &‘, d,,,,=

30

H. OSBORN

In addition (5.15) also gives the corresponding written as

version of (58a). If (2.31b) is now

then using (5.12) 2h”=

V,,kfA,,-Z(V,,k;)

A,,+2Zh==

V;,k:A4<,=

V,“,k:Ax,,,

from (5.10). Hence h” = - +(g,;k;k.‘h

+ d,;,) A,,ATh,

d:, = -d;,,

= c;,k;,.

(5.17)

Clearly (5.14), (5.16) and (5.17) verify that the gauge symmetry is preserved subject only to (5.4), (5.5). The motivation for this paper has primarily been to show how to construct string effective actions from non linear r~ models to all orders. If the dilaton field is discarded the main equations can alternatively be written as

+ (S - V’). (oi6d+

a,( w&3-l +65, --t,,S’L

(5.18)

where

from (2.33), (2.47). x plays the role of a metric on the space of. renormalisable couplings which we have calculated to two loop order, 69 . x. 69 = $cdgigjk(St,, ht,,) + ;ar2(3dC,,k, airk - 2d,, Jiik + (H2)ik

g” 67, St,, + 2&H’)”

2d,, = V,+Si,, -V,W,

6g,,),

(5.20)

+ 2H,,’ cSt,,

from (4.18) (4.26). The @cc’) term in (5.20) coincides with a recent calculation [38] of the Zamolodchikov metric on the moduli space for a general CalabiLYau D model. It is unclear as yet whether the next to leading term has any similar universal aspect (to define the metric on the moduli, dt, should be restricted to satisfy the linearised equations for vanishing 0 functions).

0 MODELS

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EFFECTIVE

ACTIONS

31

As an illustration of these results we consider the c model on squashed S’ which has SU(2) as its isometry group. This is defined by the metric [39] ds2=

i -+i, u=l cl

5, = cos $ d% + sin $ sin 8 dq5,

rs2= -sin$d%$cos$sin%d#, o,=d$+cos%db,

O,<%,(ll,

0,<$6<22n, 0<*<44n,

(5.21 )

where c’. = aNid& may be regarded as a basis of left invariant one forms on SU(2) 2 S3, (da,),= -e&~bi~cj, and the volume is 16~‘(~,&~~) --I”. The standard metric for S3 corresponds to 2, = A2 = jW3. On three dimensional manifolds, Hiik is necessarily proportional to the volume form, H,=

$V

&

Eijk,

v, H,

= 0.

(5.22)

There is in this case no global b, such that 2H,=(db), and for the quantum functional integral to be well defined, without global anomalies, we take a’=-,

4

Yf=/l,A2i3,

k

h- integer,

(5.23 )

although this is not required by the perturbative expansion valid for large k. Htlk is then independent of A, so that pfi = 0 and the space of couplings is three dimensional. If eLI are vector fields dual to co, eaio,,i = Jo,,, [e,, eh] = l UbCeC. then

1

V,e, = - EobrecLc

2

bc a ’

0

Hence in this basis for tangent vectors the Riemann tensor is constant and symmetric rank two tensors formed from g and H are diagonal so that the cr model on this manifold with the restricted space of couplings is renormalisable. Discarding the summation convention we can therefore write

Since there are no vectors even under conjugation takes the significantly simpler form

(when E,~( -+ -e&)

then (5.18)

32

H. OSBORN

In this case isI by itself, without any integration, acts as an action for the equations /3: = 0 defining a conformal field theory. The perturbative results of Section 4 may straightforwardly be specialised to this particular model. From (4.15a) and (4.26), since (4.27) gives Si = 0 to this order, 8;=fa,+~(n:+(a,-~~)2-2~2a,), (5.26) a, =22, +++!$-112’ 3

2

1

and also for /?;,3 by obvious permutations. The renormalisation flow described by these fi functions possesses an infra red stable fixed point when a, = 0 or A, = A, = A, = 1, corresponding to the SU(2) usual S3 0 model which has case, but without the torsion been considered by Gibbons

(5.27)

WZW model. If ;1, = A the model degenerates to the been discussed by Braaten et al. [40] while the general term which is necessary to provide a fixed point, has [41]. For g from (4.15a), (4.17), (4.29) we get

+~(~2(a,+a2+a3)--4(a~+a~+a~)+(a,+n,+a,)2+8q4).

(5.28)

At the fixed point (5.27) this is consistent with the expansion of

e=s=&, in accord with Virasoro central charge c = $kO for the SU(2) From (3.20), (4.27) we get C W~~=~61n~*+~(3q-l)Srl’, u

WZW

model.

(5.29)

so that at least to this order the W terms in (5.26) disappear. The metric xob may also be determined from (5.20). At lowest order it is easy to see that

x$‘=i hab,

(5.30)

(3 MODELS

which is compatible

AND

STRING

EFFECTIVE

ACTIONS

33

with (5.25), (5.28) by using I,~(~,+or,+a,)=n,-212. 1

In addition

from (5.20)

(5.31) which we have also verified is consistent with the three loop contribution also the two loop results for fib. In general in this model from (5.25) $g,=

-E,

or

to 8 and

(5.32)

which is essentially the exact result of Zamolodchikov for the renormalisation flow of his C function since from (5.30) Xnb is positive definite for weak coupling. Perhaps by extending the above considerations to general squashed group manifolds, preserving a subgroup H, it may be possible to demonstrate renormalisation flow between two fixed points within renormalisable field theories.

APPENDIX

A

Here we discuss the modifications in the treatment given in Section 2 necessary where Ed” is restricted to be strictly two dimensional, obeying (1.11) as well as (1.12) with y,,” the metric on z2. This is the traditional recipe [25] for accommodating the anti-symmetric tensor consistently while using dimensional regularisation. The required contributions for finiteness are then only restricted by invariance under reparameterisations preserving the product form on the total space z7- ; ,,-LXZ with coordinates zr = (z,, xJ. The choice of a complex basis is now 29 restricted to Z,, so that in (2.12) e: + ep, eg + ?‘. The metric on Ed may be decomposed as Y/l” = yp + yflV, jjpwy”” = 0 and it is sufficient to restrict f,,” to depend only on i,,E.@ and yrv on x,E&. For calculations it may be convenient to take g to be flat, with the topology of a torus Td-* [42], but this is not necessary. In addition to the basic Lagrangian L(I; 4) given by (2.1) the requirement of renormalisability requires that we introduce in addition

(A.1 1

34

H. OSBORN

with 98 the scalar curvature formed from fPv (note that 8 = W + 8). The full action is thus S=[

jd2x&z, E

,?=L(A)+f(R),

(A.2)

where SE denotes integration over 2 with measure d-&a&, SEl= P-* 1 as e+O. As before the couplings J in (2.2) and also the evanescent couplings 2 are allowed to have an arbitrary dependence on xfl E E2. It is convenient in (A.2) to modify the couplings A + 2, L(x) = L(A) + $a’ V?Dk, where k

=-

’

1 D

(f’)“a’&.

P 0’

so that

vpi = vpi + $x’ ai@k,)

h = - $’ (y’&Jk cc’

E=h+fi,

(A-3)

In this case if

giving 9 --t e*“(9 + 2V2a), then

This extension of Weyl symmetry of local resealings of the metrix is now a property of the full Lagrangian z and may therefore be assumed to be preserved under dimensional regularisation. Invariance under the symmetries (2.5) is also straightforward since V, 6’” = 0 and ‘yPv is independent of xP. Including all necessary contributions 2 -1, where I-, x0, given by (2.13), (2.14) and (2.20) with Y’ = 0, and 1 + 1’ = (SLY,6’). Instead of (2.16) we now take

-X,= (Apii,a;g,,

-k,’

a:H,,

Apij = - (dpp),i - ie,” &b,, to allow for possible counterterms g;=p-ygg+

(g-l)lk yxv

LQ,), (A-6)

= 0,

depending on ahi. Furthermore TV),

&=p--E(d+

!P),

(A.7)

with p$ a tensor and @ a scalar on it4 each depending only on tii and gij and containing poles in E. The definition of B functions may also now be extended to the evanescent couplings giving

CT MODELS

AND

STRING

EFFECTIVE

35

ACTIONS

Invariance under the symmetry (A.4) requires that for I-+ e2’X then f” + eZo~’ and hence also for fi” (in WZ W models when M is a group manifold there is a unique group invariant scalar product on the tangent space TM so that gii= rgU for r a constant on M and also this then implies gly = ,~‘%g~ in accord with the arguments of Bos [27]). As a consequence of S, =jE J d2x J’; 2, defining a finite theory for arbitrary xcZ2 dependent I finite local composite operators are obtained by functional differentiation just as in (2.25). In particular this gives for a scalar F [F] = pP 1 ZF, P

with SEalso present in other such similarly we may define

obtained normal products. Furthermore

giving extra finite local scalar operators. The energy momentum tensor is again as defined by (2.36) and a simple consequence of the assumption of symmetry under (A.4) is that yTp,. = 2[h] + ia v2 [@I - [Jo’&

a,(# a,.fp-j - [hi@.

(A.lO)

The derivation of Eq. (2.26a, b) and also (2.30) proceedes as before although in the latter case it is necessary to extend L(x,) -+ L(&) + &PVj). The crucial relations (2.32) and (2.35) also remain valid, although J& = 0 and it is now appropriate to take from (A.6) A=(-W,+2Qt

%gu, XJJ,,,

($-‘Jik

%g,).

(A.ll)

The conditions for the symmetry (A.4) to be preserved under renormalisation, which is necessary to obtain (A.lO), are easily seen to imply (A.12) Applying the renormalisation group equations, with the assumed structure of dependence on CJLg, in (A.3) and (A.6) which ensures that the Q, dependent part of (A.12) is automatically satisfied, this gives

oif=Bi.a=+aie, x , x . x = i a73,

9 = (0, 0, 0, Sj), .%.x.9=po,

(A.13a) (A.13b)

36

H. OSBORN

where X = epXfl, a = e” d,. The results (A.l3a, b) may then be regarded as versions of (2.48b) and (2.53) in this regularisation procedure. As before (A.13a) follows from (A.13b) by taking X + $, when $0 --f 9”0= vi ai0 and then using (2.35). To calculate with the action (A.2) using conventional techniques it is necessary to set &=

(A.14)

g,+t,,

and treat rii perturbatively. At one loop, by expanding i to order 5’ and calculating contributions of up to O(r’), this gives, after extension to a form consistent with the symmetry (A.4),

Beyond one loop there are differences as compared with the results of Section 4 with the present prescription for E,,. Replacing (4.24) we now have

As compared with (4.26) this gives Qc2)= (4~‘)’ (H2)“Viaj. For the dependence on the curvature, using also i, = + gij;kfflv V, ti V,
1 (?H

2

+ aI- gij;k g*ik;j -r 36 gij; k pk

+ O(9)).

(A.17)

Although (A.lO) defines the trace of the energy momentum tensor as a finite local operator it is manifestly of no use for identifying the conditions on the basic couplings necessary for conformal invariance of the renormalised two-dimensional theory. Clearly [.&‘I = O(E) since Zp’ is a finite operator and 8 = O(E). Hence terms involving 8 may be discarded but [f”“p$ a,#‘a,@] contains both O(E) and O(1) pieces beyond the tree level. To separate these in an appropriate manner it becomes necessary to consider general finite local composite operators of the form

C$l, a^,& WI,

(A.18)

(3 MODELS

AND

STRING

EFFECTIVE

37

ACTIONS

for any 2, with 8, the derivative with respect to f,, and vii an arbitrary symmetric tensor on M. Such operators require additional counterterms, beyond those given by q . a/agg,y and q . aJag do contributing to f 8,1$’ J,@ and i&)pv, respectively, which are proportional to frv. Under contraction with $“” finite operators such as in (A.18) are O(E) and we may then expand the resulting operator in terms of the complete set of previously defined scalar operators with dimensions d 2 as

- CW(?))l - cwbm

(A.19)

-yLwrln

in a similar fashion to Curci and Paffuti [26]. L(q), f(q), and J”(q) arise from higher order loop effects and may be computed by identifying those counterterms in (A.18) proportional to fpY/s [25]. At one loop L(A(r/)) +v,

P’(q) = $a’tj&-1)”

gqw,,+

$awk,9)

+d(~-l)ikd~((~-l)~'~~g~,-

- &-‘)“.z%

gfla;gk,)+ o((a;g)%, (A.20a)

i(J(d)=

ia’?‘” +

adk qjj:k

&~‘(~-‘)‘”

g,(,;n)

-

$,$k

~‘“h&fkpRP~mH~I+ &;,

+

. . . ) -

i $a’&j

$$t

itk(;(mn))

(A.20b)

with xkI and b, defined as in (4.9). An important check is that the rhs of (A.20a, b) should be invariant under the symmetry (A.4). By inverting p$ = qti - I”(v)~ (A.19) since the lhs is O(E), then ensures that [I&)] N [L(L(q))] +V,[.Y(r)] so that evanescent operators such as (A.9) may be eliminated. From (A.lO), using (A.19) and (A.20a) for qij= ii,, we may then write to one loop order 2 , P’“T i

Ir,, - 2[h] -;

a’[Ig”Q

-

; at (

)

(A:21 )

8 = @ - i ln( det g/g), discarding O(E) terms. This result is identical to this order with (2.56) since, from Section4, .L(p”‘)+V~U~)= $ cl’(dDW - g”w, - ~a’g~@~~9?) apart from the term involving In det g in $ which is necessary to preserve the symmetry under (A.4). However, despite these one loop results the appearance of p functions to all orders in the expression of ypvT,,, in terms of finite composite operators is no longer manifest. This may be resolved, within the philosophy of this paper, by extending the previous discussion of the quantum field theory corresponding to z = L $ i to allow for ‘y^pvalso to have an arbitrary dependence on x as well as on i. It is then necessary in (2.5) and also (A.6) to take a, -+ 8@= ap + %;l for 9; = $fup a,,~op so that se afl = d, SE and the gauge symmetries are preserved. In this case there are

38

H. OSBORN

additional possible counterterms, and corresponding evanescent couplings, necessary for finiteness. In general these may be taken to depend on the vector VP;, which may appear similarly to =?&in Section 2, and the scalar Y = i ~~~~~ Py^OP (from the full Riemann tensor &‘fi6”P formed from pP” = yP” + fp” it is then possible to form scalars of dimension 2 which can occur in counterterms yPyyuPBPPyP= 3, yp”y^op&p~“p= -v, -Y-b+ Y, y^~“y^“p~p~“p= & - “y^ Yv - Y with 8 the usual scalar curvature on g obtained from fP” by itself). In this case we may now derive a relation of the form

= 2[i(1)3

+ [i(p)]

+ [L(p^)] + V,KY

(A.22)

with K’ a finite operator. The lhs of (A.22) is O(E), since SS,,B~fi” is a finite operator, and hence this may be used in (A.lO) to obtain a finite expression for yP”TP” in terms of the j3 functions for the couplings A and 2. Unfortunately it is still necessary to use (A.19), which requires further calculations, to eliminate the evanescent operator i(B”) and so achieve an expression of the same form as (2.56) yQ”Tpv N [L(B)]

+2[h]

+ gdV[@]

+v,P,

(A.23)

with p a modified /? function, differing from /I beyond one loop, and JP arising from counterterms are given by

Ku in (A.22) and also P’ in (A.19). At one loop the necessary additional

The VP term gives rise, according to (A.22) and (A.23), to the same additional pieces in yP”TP” as are obtained in (A.21) by @ + 3. At two loops the resulting dependence on the curvature L%in (A.23) is given by fl” where from (A.15), (A.20a) this is given by

* (R + H2)k’Vka,r

1 = --a’SZ”‘iln(det 2

g/g)-

21

j (R + H2).

(A.25)

0 MODELS

AND

STRING

EFFECTIVE

39

ACTIONS

If we consider the change of variable 6@ = - $ ln(det g/g), then, with the usual corresponding

redefinition

6M=~a’(R-Bg.~)~ln(det =-a’Q(‘)‘ln(det 21 6

of /? functions [23], g/g)

g/g)+

21p-H’)+

...,

which with (A.25) recovers the earlier two loop result (4.17). By power counting there is no restriction on the dependence on the evanescent coupling iii at higher orders, apart from the symmetry (A.4). General theorems on the equivalence between different regularisation schemes guarantee that this dependence can be removed by a redefinition of the basic couplings 1, as shown above in the simplest lowest order case. However, implementing this to arbitrary order is complicated and leads to problems in deriving important results to all orders such as the Curci-Paffuti relation [26]. The essential diffkulty in this approach is that the evanescent contributions modify the leading dimension two part of the basic Lagrangian, as in (A.2), rather than, as in (2.7) for the treatment of Section 2, the lower dimension parts which then ensures that the appearance of the evanescent couplings is strongly constrained and they can be easily eliminated.

APPENDIX

B

As was discussed in the introduction, dimensional regularisation as it is usually applied violates scale invariance when this is present in the initial classical twodimensional o model and so inevitably a quantum anomaly in the trace of the energy momentum tensor arises. Nevertheless for S,(y) the usual regularised action, defined on a d dimensional space 8 and giving a finite quantum field theory, then if y”” -+ y^“”= ezVy@“,cp= In yJ2d with y”” regarded as the physical metric, invariance under Weyl resealings, y’l” + eZOy~” is trivially valid for general d, and hence is a symmetry of the renormalised quantum theory. In this case

where L, is otherwise as constructed in Section 2. In this case reparameterisation invariance on Z .IS no longer automatic. For a diffeomorphism 6x” = t”(x) with

40 then we may define, completing

H. OSBORN

(18),

Although d; S(y^)= 0, when d= 2, the action of A: on S, is non zero. The anomalous terms arise solely from the explicit dependence of S, on cp and using Af’ 40- ti ai’p = i?,,<“/d then, following a similar argument as in Section 2,

where the result is analogous to (2.56). By using the equation therefore express the action of Af’ on S, in terms of i A& for

of motion

we

where Bh=epzp Ah+ ... with the remaining terms as in (2.58). In general diffeomorphisms form a closed algebra, [A;, A;] = -Aft,,.,, and since [A;, A:] N -Agape this reproduces (1.10) using 5~a~av5’“-5’~aclaySY=a,[5, l’]” (O(E) contributions arise from divergence terms in Bh, for example, A,WV’2@ = 20 V”Qj + E Pa a;@ with similar results for A:). The crucial condition (1.6), analysed in Section 3, is therefore required for consistency once more. It is clear from the above that it is possible to maintain Weyl invariance throughout although then reparameterisation invariance is lost but this does not appear to be advantageous in actual calculations [24]. APPENDIX

C

In order to undertake perturbative calculations using dimensional regularisation with an arbitrary curved background metric while maintaining manifest covariance for general d-2 we use the straightforward modification to two dimensions of techniques developed some time ago for similar multi-loop calculations in four dimensional field theories [43]. This method depends crucially on an expansion of the Green function of the form (4.21) where G,(x, x’), G,(x, x’) are given in terms of the biscalar (T(x, x’), satisfying a,0 Pa = 2a and at coincident points

CY MODELS

AND

STRING

EFFECTIVE

41

ACTIONS

v,a,d = gyvy and d(x, x’) where AV2a+ 2a’*iJ,A = dA, Al = 1 [34]. They are effectively defined by the equations, with p an arbitrary

scale mass,

(-V2+(A-‘V2A))G,=G,,-%A, (-$,+(A-‘a,A))G,=

+,oG,,.

For general x, x’ GO, G, are regular as d -+ 2 but they also have well defined coincidence limits G,( =G,

-V’G,,

=Gd.

(C.2)

The divergences of loop amplitudes for two dimensional Q models arise from the singular parts in the products of GO’s and G,‘s with up to four derivatives. We give here the relevant formulae for arbitrary products of G,‘s, with also one G,, involving just two points x, x’. The singular parts are then given by an appropriately symmetric covariant tensorial form involving derivatives of 6”(x, x’) with the number of derivatives determined by power counting. It is easy to derive (C.3a)

(C.3b) V,V,,Go V,G, GI;- ’ --- 1 1 p-nB 2dYr,,a~-Y,~a,,-“i,,,ap - -2+dn (27~s)”

>

sd,

(C.3c)

V,Go V,Go V,G,G”,-2 1 p-nE z--- E d(2 + d) n(n - 1) (271~)~(Ye,,a, + ylrrr4 + Y”, a,,) sd.

(C3d)

The result (C.3a) is justified by contraction with yPv and using (C.l), (C.2) while (C.3b) is obtained from this by discarding a non singular total derivative and (C.~C, d) are then calculated from the derivatives of (C.3a, b).

42

H. OSBORN

For four derivatives arguments V,VJ’,G

we may therefore derive more

laboriously

by similar

G;; 1

pL-“&

(Y,, V” v, + YPQv, v, - - d+ 4 (27&s)” i + Y "0 v, v, + Y vp v, v, + Yfl" v, v, + ygp v, V,) dd -~(YY.YI~+YP~Y"P+Y~PY"")v26d

V&G,,

I

V,V,G,G;;-’ 1 j.P& - -

1

w(d+2)(d+4)n(2m)n

d(y,m Vv V, + ~pp Vv V,

+ Y”, v, v, + Yvpv, V,) ad - 4(Y,” v, v, + Yapv, V”) dd

1 +(d+2)(d+4)

n-l --n

pP= 1 (2m)“6

(C.4b)

d MODELS

E - (d+ 2)(d+

AND

STRING

EFFECTIVE

ACTIONS

43

1 ,lP= ~~ (Y,J”v,+Y,,vx 4) n(n - 1) (27cs)” i

(C.4c)

V,GoV,GJ,G,V,GoG'd-3 - nt: P __ P(Y,, (2~~3”

1 l)(n-2)

E2

-d(d+2)(d+4)n(n-

+YllpV”vc+Y"0 v,

v, + Yvp v,, v, + YP" v, v, + Yol, v,, V,,) s"

+ (Yp7Y"P + YrpYYo + Yfl"YOJ v2q E

-“E

1

P

1

+ d(d+ 2) n(n - l)(n - 2) -(27E)” -3 U~‘,vYyp

E

V”v,

1

,Knz

+d(d+2)(d+4)n(nx; b~(YpvYop + Y,mYvp+ YgpYlr)- 4(d+ 1)

44

H. OSBORN

It is also necessary to consider products involving G,. Any derivative of G, can be reduced to products of G, by virtue of (C.l). For products containing G, and four derivatives the singular parts are proportional to 6 functions without any derivatives or curvature tensor terms. The main results are

G,V,V,V,V,G&-' (C.5a)

2 S”, (C5b) ~Y~YYQ~-Y~dYY~-Y~~YYu

G,V,V,G&,GoV,GoG;;-

3

sd. ~YrYYop-YpcoYvp-Y 2

(C.5c)

The formulae (C.3), (C.4), (C.5) for the singular parts of products of G,‘s and also G1 are given only for derivatives on GO with respect to X. They may be extended to allow for derivatives with respect to x’ by introducing yPy’, the matrix giving parallel transport of vectors along the geodesic from x’ to x [34]. (C.3) and (C.5) remain valid if on the lhs we take v

o -, ypCY’G,fjv,,

CC.61

while in (C.4b, c) it is sufficient to take

+ ---1

1 ,LPE

d+2n

---

E

1 L@pvYap+ se,,Y,“) - %pw -~wTvp d

(2m)”

1

pPs __

84

1 - GqIYYBP + ~ppyo + ~wp) Sd9

d(d+ 2) n(n - 1) (27~)” 2

(C.7a)

(C.7b)

using V,V,Y,~’ 1= - +Wp’rrPy. In each of the above expressions functions of d should be expanded keeping only poles in E on the rhs. The different prescriptions for treating E,, lead to very different results beyond lowest order. With E,, d-dimensional as in Section 2 then from (C.4b, c) and (C7a, b), and using relations such as XI = d, cY,.k’l= 0, V*Xl = E,, V*&‘” when X= •~“y~~‘y~~‘~~,~., we obtain

0 MODELS

AND

STRING

EFFECTIVE

ACTIONS

45

(CXa)

-N -2d(d+2)n(n-1)(2m)”

(C8b) However, for E,, 2-dimensional we may use, subject to the conditions paragraph of Appendix A, E,, = Y,,~‘Y~~‘E~,~,and also (1.11) to give 6”‘~~‘“’ V,G,V’,. V,,G&
1 /.-“c

d + 2 n (27~)”

of the first

’

-2V’+bp2+3?

+(;(d+l)W-ii%

(C.9a)

E~“E”‘~’ V,G,V,. V,G,G,V,G;-

*

1 N--d(d+ 2) n(n - 1) (27~)” +(n-1)

(C.9b)

where q2=V2+V2 and &?=$$a&. For the purpose of two loop calculations as in Section 4 we need the singular parts after subtraction of subdivergences according to the general Bogohubov recursion formula. Using (C.ga, b) Y(E~“E~‘~’ V,G,V,. V,G&G,) = 6”“~“‘~’ V,G&

V,G&.

G,

(ClOa)

46

H. OSBORN

(C.lOb) With the alternative treatment

of (C.9a, b)

9’(~pv~o’P’ V,G,,V,r V,G&

G,) (C.lla)

Y(E”“E~‘~’ &Go V’,, V, G,G&) -2 Similarly

for d-dimensional

1 -p2+82)-+-;~}6d, (471s) i

(C.llb)

E,,

~‘(E”“E”‘~’ V, G,V,. V,G& - ~(E”Y’P’G,

G,)

V, G&.

V, G&J

- - 9’( E”“E~‘~’ V, G rV’,r V, Go G,V,) - - Y(E~“E~‘~’ V,G&

- -&l

V,G, G,,V,)

-&)hd,

(C.12)

while for the 2-dimensional epv, 1 --E + 1 - 4s. The results (ClOa, b) and (C.12) were used to obtain (4.24) and correspondingly for (A.16). Finally in order to obtain (A.24) it is necessary to be able to compute with the perturbation t’ = $‘“rii a,,@ 13,@ to order O(?) at one loop for general fP,,(x, a). The crucial ingredient, with the definitions of Appendix A, is y^pvf=‘p’V,G,~,r

N&

V, G&

{ -Q2dd+

ACKNOWLEDGMENT I am grateful

to Ian Jack for many

very

helpful

conversations.

(V, Y” + “$V”)

dd}.

(C.13)

(3 MODELS

AND

STRING

EFFECTIVE

ACTIONS

47

REFERENCES

1. C. LOVELACE, 2. E. S. FRADKIN 3. C. G. CALLAN, 4. 5. 6. 7.

8. 9. 10. 11. 12.

13. 14.

15.

16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

Pbys. Lets. B 135 (1984), 75; Nucl. Phys. B 293 (1986), 413. AND A. A. TSEYTLIN, Phys. Left. B 158 (1986), 316; Nucl. Phys. E 261 (1985), I. D. FRIEDAN, E. J. MARTINEC, AND M. J. PERRY, Nucl. Phys. B 262 (1985), 593; C. G. CALLAN, I. R. KLEBANOV, AND M. 1. PERRY, Nuct. Phys. B 278 (1986j, 78. A. SEN, Phys. Rev. L&t. 55 (1985), 1846; Phys. Rev. D 32 (1985), 2102. A. A. TSEYTLIN, Int. J. Mod. Phys. A 4 (1989) 1257. D. FRIEDAN, Ann. Phys. (N.Y.) 163 (1985). 318. T. BANKS, D. NEMESCHANSKY, AND A. SEN, Nucl. Phys. E 277 (1986), 67; A. DAS AND S. Roy. 2. Phys. C 36 (1987), 317; K. KIKKAWA, M. MAENO, AND S. SAWADA, Phys. Lett. B 197 (1987). 324; M. MAENO AND S. SAWADA, Nucl. Phys. B 306 (1987). 603; A. DAS, Phys. L&f. B 205 (1988). 49; Y.-X. CHENG, Phys. Let!. E 218 (1989), 39. A. B. ZAMOLODCHIKOV, JETP Left. 43 (1986), 730; Sov. J. Nucl. Phys. (Engl. Trawl.) 46 (1987). 1090. A. A. TSEYTLIN, Phys. Let?. B 178 (1987), 34; Nucl. Phw. E 294 (1987). 383. G. M. SHORE, Nucl. Phys. B 286 (1987), 349. A. A. TSEYTLIN, Phys. Left. B 194 (1987), 63. B. E. FRIDLING AND A. E. M. VAN DE VEN, Nucl. Phys. B 268 (1986), 719; E. GUADAGNINI AW M. MINTCHEV. Phys. Left. 186 (1987), 173; “Beta Functions and Central Charge of Generalised u Models,” preprint IFUP-TH 21/86; I. JACK AND D. A. Ross, “Renormalization of a TwoDimensional Supersymmetric Model in a General Background Field,” Southampton preprint SHEP 86/87-9; D. ZANON, Phys. Letr. B 191 (1987), 363; D. R. T. JONES, Phys. Lett. B 192 (1987). 391; S. V. KETOV, Nuci. Phys. B 294 (1987), 813; JETP Lair. 47 (1988). 339; C. M. HUI.L AND P. K. TOWNSEND, Phys. Letl. B 191 (1988), 115. R. R. MET~AEV AND A. A. TSEYTLIN, Phys. Left. B 191 (1987), 354; Nucl. Phys. E 293 (1987), 385. S. J. GRAHAM, Phys. Letr. B 197 (1987) 543; A. P. FOAKES AND N. MOHAMMEDI, Phys. Left. E (1987), 359; Nucl. Phys. B 306 (1988), 343; I. JACK, D. R. T. JONES, AND N. MOHAMMEDI, Phw. Left. B 220 (1989), 136; Nucl. Phys. E 322 (1989). 431. C. NUNEZ, Phys. Lett. E 193 (1987), 195; I. JACK AND D. R. T. JONES, Phys. Left. B 193 (1987). 449; S. BELLUCCI, Prog. Theor. Phys. 79 (1988), 1288; 2. Phys. C 41 (1989) 631; M. C. BENTO, 0. BERTOLAMI. A. B. HENRIQUES. AND J. C. ROMAO, Phys. Left. 218 (1989). 162; Phw Let!. 220 (1989), 113. I. JACK AND D. R. T. JONES, Phys. Letr. E 200 (1988). 453. C. M. HULL AND P. K. TOWNSEND, Nucl. Phys. B 301 (1988), 197. G. CURCI AND G. PAFFUTI, Nucl. Phys. E 286 (1987), 399. D. FRIEDAN, in “Recent Advances in Field Theory and Statistical Mechanics.” (J.-B. Zuber and R. Stora, Eds.) North-Holland, Amsterdam, 1984. S. P. DE ALWIS, Phys. Rev. D 34 (1986), 3760. I. JACK, D. R. T. JONES, AND D. A. Ross, Nd. Phys. E 307 (1989), 531. H. OSEiORN, Nucl. Phys. 8294 (1987), 595. H. OSBORN, Nucl. Phys. E 308 (1988), 629. E. GUADAGNINI, “Interacting Physical States of the Bosonic String I and II,” preprints IFUP-TH 27/W, 28/88. G. BONNEALJ, “Renormalization, Regularization. a critical but non-technical review,” preprint PAR LPTHE 88-52. G. CURCI AND G. PAFRJTI, Nucl. Phys. B 312 (1989). 227. M. Bos, Phys. Lett. E 189 (1987), 435; Ann. Phys. (N.Y.) 181 (1988), 177; Z.-M. XI, Phys. Letr. B 214 (1988). 204; Nucl. Phys. E 314 (1989), 112. J. C. COLLINS, “Renormalization,” p, 345, Cambridge Univ. Press, London, 1984. G. BONNEAU, Nucl. Phys. B 167 (1980), 261; B 171 (1980), 477; fhys. Left. B 96 (1980). 147; NW~. Phys. E 177 (1981), 523.

48

H. OSBORN

30. K. YANO, “Differential Geometry on Complex and Almost Complex Spaces,” Pergamon, Elmsford, N.Y., 1965. 31. M. T. GRISARU AND D. ZANON, Nucl. Phys. B 252 (1985), 578; M. T. GRISARU, B. MILEWSKI, AND D. ZANON, Phys. Letr. B 155 (1985), 357; 157 (19SS), 174; Nucl. Phys. 266 (1986), 589. 32. H. OSBORN, Phys. Lett. B 214 (1988), 555. 33. S. MUKHI, Phys. Left. B 162 (1985), 345; Nucl. Phys. B 264 (1986) 640. 34. B. S. DEWITT, “Dynamical Theory of Groups and Fields,” Gordon & Breach, New York, 1965. 35. P. S. HOWE, G. PAPADOPOULOS, AND K. S. SIXLLE, Nucl. Phys. B 296 (1987), 26. 36. R. I. NEPOMECHIE, Phys. Left. B 171 (1986), 195; Phys. Reo. D 33 (1986) 3670; E. GUADAGNINI, M. MARTELLLINI, AND M. MINTCHEV, Phys. Lat. B 194 (1987), 69; E. GUADAGNINI, Nucl. Phys. B 290 (FS20) (1987), 417; L. S. BROWN AND R. I. NEPOMECHIE, Phys. Rev. D 35 (1987) 3239; L. S. BROWN, G. J. GOLDBERG, C. P. RIM, AND R. I. NEPOMECHIE, Phys. Rev. D 36 (1987) 551; D. KARABALI, Q.-H. PARK, H. J. SCHNITZER, AND Z. YANG, Phys. Let?. B 216 (1989), 307; D. KARABALI AND H. J. SCHNITZER, Nucl. Phys. B 329 (1990) 649; P. BOWCOCK, Nucl. Phys. B 316 (1989), 80; K. GAWEDZKI AND A. KUPIANEN, Phys. Lefr. B 215 (1989), 119; Nucl. Phys. B 316 (1989), 625. 37. I. JACK, D. R. T. JONES, N. MOHAMMEDI, AND H. OSBORN, “Gauging the General o-model with a Wess-Zumino Term,” Nucl. Phys., in press; C. M. HULL AND B. SPENCE, Phvs. Lett. B 232 (1989). 204. 38. P. CANDELAS, P. S. GREEN, AND T. H~BSCH, Phys. Rev. Left. 62 (1989), 1956; Nucl. Phys. B 330 (1990), 49; P. CANDELAS, T. H~~BSCH, AND R. SCHIMMRIGK, Nucl. Phys. B 329 (1990), 649. 39. M. J. DUFF, B. E. W. NILSSON, AND C. N. POPE, Phys. Rep. 130 (1986), 1. 40. E. BRAATEN, J. L. CURTWRIGHT, AND C. K. ZACHOS, Nucl. Phys. B 260 (1985), 630. 41. G. W. GIBBONS, “The Ricci Flow for Left Invariant Metrics on SO(3) gives S0(3)-invariant Self-dual 4-metrics,” Cambridge preprint. 42. M. L~SCHER, Ann. Phys. (N. Y.) 142 (1982), 359. 43. I. JACK AND H. OSBORN, Nucf. Phys. B 234 (1984). 331.